Injection Campaigns and Search Sensitivity

An injection campaign measures what a frozen search recovers, not merely what signals are present. Simulated waveforms are placed in representative detector data, passed through the same conditioning, ranking, veto, and threshold logic as foreground data, and used to estimate efficiency over source parameters. This page derives sensitive spacetime volume as an importance-sampling integral, validates the estimator against an analytic toy problem, and states the data, calibration, software, and selection information needed to interpret a reported sensitivity.

Required background. Probability and Statistical Inference supplies Bernoulli experiments, Monte Carlo estimators, importance weights, and uncertainty propagation.

Helpful background. Gravitational Radiation and Waveforms supplies injected signal models, while Computational Relativity supplies numerical waveforms and convergence tests.

Efficiency is conditional

Let $\vartheta$ collect source-frame masses, spins, redshift, sky position, inclination, polarization, and event time. For a fixed pipeline and detection rule $D$, define

$$\epsilon(\vartheta) =P\!\left(D=1\mid\vartheta,\mathcal A\right),$$

where $\mathcal A$ denotes the detector state, calibration, conditioning, bank, search version, category, and threshold. The probability averages over noise realizations and any stochastic pipeline elements. It is not an intrinsic property of the source alone.

For sources distributed in detector-frame time and comoving volume, the sensitive spacetime volume for an intrinsic population density $p_\mathrm{pop}(\xi)$ is schematically

$$\langle VT\rangle =\int_{\mathcal T}dt \int dz\,d\xi\,d\Omega\,d\iota\,d\psi\, \frac{dV_c}{dz}\frac{1}{1+z}\, p_\mathrm{pop}(\xi)\,p(\Omega,\iota,\psi)\, \epsilon(\vartheta).$$

Here $dV_c/dz$ is the full-sky comoving-volume element, $p(\Omega,\iota,\psi)$ is normalized over direction and orientation, and $t$ is detector-frame time. Thus no second factor of $4\pi$ belongs in the angular integral. The factor $(1+z)^{-1}$ converts a rate per source-frame time to expected events per detector-frame time: multiplying a constant comoving source-frame rate density by $\langle VT\rangle$ gives the expected detected count. Redshift evolution of that rate or population must be included explicitly. A quoted $\langle VT\rangle$ is conditional on these measures, cosmology, waveform family, detector segments, and threshold.

Importance sampling an injection set

Suppose injections are drawn from a normalized proposal density $q(\vartheta)$ with respect to the coordinate measure $d\vartheta$. Write the target measure as $w(\vartheta)d\vartheta$. If $I_i$ is one when injection $i$ is found according to the predeclared association and threshold rule, then

$$\widehat{\langle VT\rangle} =\frac{1}{N}\sum_{i=1}^{N} \frac{w(\vartheta_i)}{q(\vartheta_i)}I_i.$$

This is unbiased when $q$ has support wherever $w(\vartheta)\epsilon(\vartheta)$ is nonzero and the injected trials represent the target detector time. The Monte Carlo standard error is estimated from the sample variance of the weighted recovery values divided by $\sqrt N$. A few extreme weights reduce effective sample size and can make the central-limit error unreliable.

If injection sets are stratified by epoch, source class, or distance, each stratum needs its actual drawing density and allocation probability. Treating a deliberately nonphysical injection distribution as the astrophysical prior biases sensitivity. FINDCHIRP describes software-injection efficiency and effective-distance sensitivity measurements for a matched-filter pipeline in Allen et al. 2012, §§VII–VIII, PDF.

An analytic sensitive-volume benchmark

Choose a length scale $r_0$ and measure every radius below in units of $r_0$. Consider an isotropic Euclidean population inside dimensionless radius $R=3$ with a smooth detection probability

$$\epsilon(r)=e^{-r^3}.$$

The exact sensitive volume is

$$V_\mathrm{sens} =4\pi\int_0^3 r^2e^{-r^3}\,dr =\frac{4\pi}{3}\left(1-e^{-27}\right) =4.1887902048\ldots .$$

The displayed volume is in units of $r_0^3$; in physical coordinates the efficiency is $\exp[-(r_{\rm physical}/r_0)^3]$, so its exponential has a dimensionless argument.

Draw points uniformly in the enclosing volume, so

$$q(r)=\frac{3r^2}{R^3}, \qquad V_\mathrm{tot}=\frac{4\pi R^3}{3}.$$

For this controlled calculation, use the exact recovery probability rather than a Bernoulli draw. The estimator becomes $V_\mathrm{tot}N^{-1}\sum_i e^{-r_i^3}$. With Python 3.9.6, NumPy==2.0.2, and seed 161803 reset for each row, the executed results are:

Injections	Volume result	Monte Carlo standard error	Standardized error
10000	4.038889	0.144650	-1.036
100000	4.216880	0.046999	0.598
1000000	4.203417	0.014849	0.985

The error decreases approximately as $N^{-1/2}$, and every estimate is within about $1.1$ quoted standard errors of the analytic value. This validates the measure, normalization, random transform, and uncertainty calculation. It does not validate any gravitational waveform or detector pipeline.

Executable rerun

import math
import numpy as np

radius = 3.0
exact = 4 * math.pi * (1 - math.exp(-radius**3)) / 3
total_volume = 4 * math.pi * radius**3 / 3

print(f"exact={exact:.10f}")
for size in (10_000, 100_000, 1_000_000):
    rng = np.random.default_rng(161803)
    r = radius * rng.random(size) ** (1 / 3)
    contribution = total_volume * np.exp(-r**3)
    estimate = np.mean(contribution)
    standard_error = np.std(contribution, ddof=1) / np.sqrt(size)
    standardized = (estimate - exact) / standard_error
    print(size, f"{estimate:.6f}", f"{standard_error:.6f}",
          f"{standardized:.3f}")

Save the program as sensitive_volume_check.py and run:

python3 -m venv .venv
. .venv/bin/activate
python3 -m pip install numpy==2.0.2
python3 sensitive_volume_check.py

A genuine injection campaign replaces the analytic efficiency by recovered indicators and should report convergence against injection count, weight truncation or stratification, source bins, detector epoch, and independent injection sets. A deterministic seed makes the control repeatable; uncertainty still describes the sampling design, not merely repeatability of pseudorandom output.

Software and hardware injections

A software injection adds a simulated calibrated strain waveform to recorded data, usually after specifying detector response and time delays. It efficiently samples large parameter sets and can test the complete analysis downstream of the insertion point. A hardware injection actuates the interferometer control system and tests more of the sensing, calibration, acquisition, and analysis chain, but is sparse and tied to particular detector states.

Neither automatically represents the astrophysical population. Waveform accuracy, calibration uncertainty, finite sampling, signal overlaps, gating, data-quality masks, and association windows can change recovery. Injections placed only in unusually clean times overestimate average sensitivity; injections vetoed by construction but omitted from the denominator do the same. The denominator must follow a predeclared opportunity rule.

At the evidence cutoff of 2026-08-18, the GWTC-5.0 methods paper documents the versioned O4b search configurations, injection families, thresholds, and sensitivity estimation LVK Collaboration 2026, §§2–4, PDF. Those observational results are conditional on that release and analysis. This page does not reuse their injection files or quote a catalog-sensitive volume.

Reproduction and systematic checks

An interpretable campaign records the public or controlled dataset version, detector GPS segments, calibration and uncertainty realization, quality masks, sample rate, waveform approximant and code version, cosmology, source-coordinate convention, proposal density including bounds and mixture weights, injection times, random seeds, pipeline and bank commit, recovery association window, clustering rule, ranking threshold, category map, foreground exposure, and every excluded injection with reason. Retain the injection table and recovery table with hashes.

The primary result should include weighted estimate, Monte Carlo uncertainty, effective sample size, binwise efficiency, and convergence under larger or independent injection sets. Systematic variants should cover calibration response, waveform family, PSD and conditioning, population reweighting, detector epoch, threshold, and recovery logic. Failures include missing proposal support, silent changes of prior coordinates, extreme importance weights, reused noise that understates variance, injections contaminating background estimation, and threshold tuning on the same recovered sample used to report sensitivity.

Check your understanding

In the toy model above, what fraction of the enclosing geometric volume is sensitive? If injections are instead drawn uniformly in radius, $q_r(r)=1/3$, what weight multiplies $\epsilon(r)$ in the Monte Carlo average?

Solution

The sensitive fraction is

$$\frac{V_\mathrm{sens}}{V_\mathrm{tot}} =\frac{1-e^{-27}}{27} =0.0370370370\ldots .$$

For a radius-uniform proposal, the target volume element is $4\pi r^2dr$, so importance sampling gives

$$V_\mathrm{sens} =\mathbb E_{q_r}\!\left[ \frac{4\pi r^2}{q_r(r)}\,\epsilon(r) \right] =\mathbb E_{q_r}\!\left[12\pi r^2e^{-r^3}\right].$$

Thus the per-injection contribution is $12\pi r_i^2e^{-r_i^3}$. Using the enclosing volume as a constant weight would be correct for volume-uniform draws, not radius-uniform draws.

References

• Allen, Bruce, et al. “FINDCHIRP: An Algorithm for Detection of Gravitational Waves from Inspiraling Compact Binaries.” Physical Review D 85 (2012): 122006. DOI. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. “GWTC-5.0: Methods for Identifying and Characterizing Gravitational-Wave Transients.” arXiv:2605.27224 [gr-qc] (2026). Open PDF.